Inverse eigenvalue problem for a class of Dirac operators with discontinuous coefficient

In this paper, the inverse problem of recovering the coefficient of a Dirac operator is studied from the sequences of eigenvalues and normalizing numbers. The theorem on the necessary and sufficient conditions for the solvability of this inverse problem is proved and a solution algorithm of the inverse problem is given.

MSC:34A55, 34L40.

In this paper, we consider the boundary value problem generated by the system of Dirac equations on the finite interval :

with boundary conditions

where

p(x), q(x) are real valued functions, p(x)\in L_2(0,\pi ), q(x)\in L_2(0,\pi ), λ is a spectral parameter,

and 1\ne \alpha >0.

The inverse problem for the Dirac operator with separable boundary conditions was completely solved by two spectra in [1, 2]. The reconstruction of the potential from one spectrum and norming constants was investigated in [3]. For the Dirac operator, the inverse periodic and antiperiodic boundary value problems were given in [4–6]. Using the Weyl-Titschmarsh function, the direct and inverse problems for a Dirac type-system were developed in [7, 8]. Uniqueness of the inverse problem for the Dirac operator with a discontinuous coefficient by the Weyl function was studied in [9] and discontinuity conditions inside an interval were worked out in [10, 11]. The inverse problem for weighted Dirac equations was obtained in [12]. The reconstruction of the potential by the spectral function was given in [13]. For the Dirac operator with peculiarity, the inverse problem was found in [14]. Inverse nodal problems for the Dirac operator were examined in [15, 16]. In the case of potentials that belong entrywise to L_p(0,1), for some p\in [1,\mathrm{\infty }), the inverse spectral problem for the Dirac operator was studied in [17], and in this work, not only the Gelfand-Levitan-Marchenko method but also the Krein method [18] was used. In the positive half line, the inverse scattering problem for the Dirac operator with discontinuous coefficient was analyzed in [19]. Besides, in a finite interval, for Sturm-Liouville operator inverse problem has widely been developed (see [20–22]). The inverse problem of the Sturm-Liouville operator with discontinuous coefficient was worked out in [23, 24] and discontinuous conditions inside an interval were obtained in [25]. In the mathematical and physical literature, the direct and inverse problems for the Dirac operator are widespread, so there are numerous investigations as regards the Dirac operator. Therefore, we can mention the studies concerned with a discontinuity, which is close to our topic, in the references list.

In this paper, our aim is to solve the inverse problem for the Dirac operator with a piecewise continuous coefficient on a finite interval. Let {\lambda }_n and {\alpha }_n (n\in \mathbb{Z}) be, respectively, eigenvalues and normalizing numbers of the boundary value problem (1), (2). The quantities \{{\lambda }_n,{\alpha }_n\} (n\in \mathbb{Z}) are called spectral data. We can state the inverse problem for a system of Dirac equations in the following way: knowing the spectral data \{{\lambda }_n,{\alpha }_n\} (n\in \mathbb{Z}) to indicate a method of determining the potential \mathrm{\Omega }(x) and to find necessary and sufficient conditions for \{{\lambda }_n,{\alpha }_n\} (n\in \mathbb{Z}) to be the spectral data of a problem (1), (2). In this paper, this problem is completely solved.

We give a brief account of the contents of this paper in the following section.

Let S(x,\lambda ) be solution of the system (1) satisfying the initial conditions

The solution S(x,\lambda ) has an integral representation [26] as follows:

where

A={(A_{ij})}_{i,j=1}^2 is a quadratic matrix function A_{ij}(x,\cdot )\in L_2(0,\pi ) and A(x,t) is the solution of the problem

Equation (4) gives the relation between the kernel A(x,t) and the coefficient \mathrm{\Omega }(x) of (1). Let \psi (x,\lambda ) be solutions of the system (1) satisfying the initial conditions

The characteristic function \mathrm{\Delta }(\lambda ) of the problem (1), (2) is

where W[S(x,\lambda ),\psi (x,\lambda )] is the Wronskian of the solutions S(x,\lambda ) and \psi (x,\lambda ) and independent of x\in [0,\pi ]. The zeros {\lambda }_n of the characteristic function coincide with the eigenvalues of the boundary value problem (1), (2). The functions S(x,\lambda ) and \psi (x,\lambda ) are eigenfunctions and there exists a sequence {\beta }_n such that

Denote the normalizing numbers by

The following relation is valid:

where \dot{\mathrm{\Delta }}(\lambda )=\frac{d}{d\lambda }\mathrm{\Delta }(\lambda ). In fact, since S(x,\lambda ) and \psi (x,\lambda ) are solutions of the problem (1), (2), we get

Multiplying the equations by S_1^{\prime }(x,{\lambda }_n), S_2^{\prime }(x,{\lambda }_n), -{\psi }_1^{\prime }(x,\lambda ), -{\psi }_2^{\prime }(x,\lambda ), respectively, adding them together, integrating from 0 to π and using the condition (2),

is found. From (6) as \lambda \to {\lambda }_n, we obtain

The following two theorems are obtained by Huseynov and Latifova in [27].

Theorem 1 (i) The boundary value problem (1), (2) has a countable set of simple eigenvalues {\lambda }_n (n\in \mathbb{Z}) where

1. (ii)

   The eigen vector-functions of problem (1), (2) can be represented in the form
2. (iii)

   The normalizing numbers of problem (1), (2) have the form

   {\alpha }_n=\alpha \pi -\alpha a+a+{\delta }_n,\{{\delta }_n\}\in l_2.

   (9)

Theorem 2 (i) The system of eigen vector-functions \{S(x,{\lambda }_n)\} (n\in \mathbb{Z}) of problem (1), (2) is complete in space L_{2,\rho }(0,\pi ;{\mathbb{C}}^2).

(ii) Let f(x) be an absolutely continuous vector-function on the segment [0,\pi ] and f_1(0)=f_1(\pi )=0. Then

moreover, the series converges uniformly with respect to x\in [0,\pi ].

(iii) For f(x)\in L_{2,\rho }(0,\pi ;{\mathbb{C}}^2) series (10) converges in L_{2,\rho }(0,\pi ;{\mathbb{C}}^2); moreover, the Parseval equality holds:

From [27], the following inequality holds:

where C_{\delta } is a positive number and this inequality is valid in the domain

where {\lambda }_n^0=\frac{n\pi }{\mu (\pi )} (n\in \mathbb{Z}) are zeros of the function {\mathrm{\Delta }}_0(\lambda )=sin\lambda \mu (\pi ) and δ is a sufficiently small number.

In Section 3, the fundamental equation

is derived by using the method by Gelfand-Levitan-Marchenko, where

and

In Section 4, we show that the fundamental equation has a unique solution A(x,t) and the boundary value problem (1), (2) can be uniquely determined from the spectral data. In Section 5, the result is obtained from Lemma 6 that the function S(x,\lambda ) defined by (3) satisfies the equation

where

where A(x,t) is the solution of the fundamental equation. In Lemma 7, using the fundamental equation, the Parseval equality

is found. We demonstrate by using Lemma 6, Lemma 9, and Lemma 10 that \{{\lambda }_n,{\alpha }_n\} (n\in \mathbb{Z}) are spectral data of the boundary value problem (1), (2). Then necessary and sufficient conditions for the solvability of problem (1), (2) are obtained in Theorem 11. Finally, we give an algorithm of the construction of the function \mathrm{\Omega }(x) by the spectral data \{{\lambda }_n,{\alpha }_n\} (n\in \mathbb{Z}).

Note that throughout this paper, \tilde{\varphi } denotes the transposed matrix of ϕ.

Theorem 3 For each fixed x\in (0,\pi ] the kernel A(x,t) from the representation (3) satisfies the following equation:

where

and

where {\lambda }_n^0 and {\alpha }_n^0 are, respectively, eigenvalues and normalizing numbers of the boundary value problem (1), (2) when \mathrm{\Omega }(x)\equiv 0.

Proof According to (3) we have

It follows from (3) and (16) that

and

Using the last two equalities, we obtain

or

where

It is easily found by using (14) and (15) that

Let f(x)\in AC[0,\pi ]. Then according to the expansion formula (10) in Theorem 2, we obtain uniformly on x\in [0,\pi ]

From (18), we find

It follows from (3) that

Taking into account (21) and expansion formula (10) in Theorem 2, we get

Substituting \frac{\xi }{\alpha }+a-\frac{a}{\alpha }\to {\xi }^{\prime }, we obtain

Now, we calculate

Using (7) and the residue theorem, we get

where {\mathrm{\Gamma }}_N=\{\lambda :|\lambda |={\lambda }_N^0+\frac{\pi }{2\mu (\pi )}\} is oriented counter-clockwise, N is a sufficiently large number. Taking into account the asymptotic formulas as |\lambda |\to \mathrm{\infty }

and the relations ([20], Lemma 1.3.1)

it follows from (12) and (24) that

Thus, using (17), (19), (20), (22) (23), and (25), we find

Since f(x) can be chosen arbitrarily,

is obtained. □

Lemma 4 For each fixed x\in (0,\pi ], (13) has a unique solution A(x,\cdot )\in L_2(0,\mu (x)).

Proof When , (13) can be rewritten as

where

Now, we shall prove that L_x is invertible, i.e. has a bounded inverse in L_2(0,\pi ).

Consider the equation (L_{x}f)(t)=\phi (t), \phi (t)\in L_2(0,\pi ;{\mathbb{C}}^2). From this and (24),

We show that

In fact,

Thus, the operator L_x is invertible in L_2(0,\pi ). Therefore the fundamental equation (13) is equivalent to

and L_x^{-1}{K}_x is completely continuous in L_2(0,\pi ). Then it is sufficient to prove that the equation

has only the trivial solution g(t)=0. Let g(t) be a non-trivial solution of (27). Then

It follows from (14) that

Using (21), we get

Substituting \frac{\xi }{\alpha }+a-\frac{a}{\alpha }\to \xi into the last two integrals, we obtain

Using the Parseval equality,

it follows from (28) that